Sublattice

Remarks

A sublattice $M$ is a lattice with the restricted operations, and its order is the order of $L$ restricted to $M$. Some authors allow the empty subset as a sublattice. Example: in the power set of $\set{1, 2, 3}$ ordered by $\subseteq$, the set $M = \set{\emptyset, \set{1}, \set{2}, \set{1, 2}}$ is a sublattice. The union and the intersection of two members of $M$ are subsets of $\set{1, 2}$, and all subsets of $\set{1, 2}$ are in $M$. Non-example: $N = \set{\emptyset, \set{1}, \set{2}, \set{1, 2, 3}}$ is a lattice in the induced order, where the join of $\set{1}$ and $\set{2}$ is $\set{1, 2, 3}$. It is not a sublattice, because the join of $\set{1}$ and $\set{2}$ in the power set is $\set{1, 2} \notin N$. Every chain in a lattice is a sublattice, since $x \vee y$ and $x \wedge y$ are $x$ or $y$ when $x$ and $y$ are comparable.